m 1v 10+m 2v 20=m 1v 11+m 2v 21m_1 v_{10} + m_2 v_{20} = m_1 v_{11} + m_2 v_{21}
12m 1v 10 2+12m 2v 20 2=12m 1v 11 2+12m 2v 21 2\frac{1}{2} m_1 v_{10}^2 + \frac{1}{2} m_2 v_{20}^2 = \frac{1}{2} m_1 v_{11}^2 + \frac{1}{2} m_2 v_{21}^2
v 21=m 1v 10+m 2v 20−m 1v 11m 2v_{21} = \frac{m_1 v_{10} + m_2 v_{20} - m_1 v_{11}}{m_2}
m 1v 10 2+m 2v 20 2=m 1v 11 2+m 2(m 1v 10+m 2v 20−m 1v 11m 2) 2 =m 1v 11 2+m 2((m 1v 10+m 2v 20m 2) 2−2m 1(m 1v 10+m 2v 20)v 11m 2 2+m 1 2m 2 2v 11 2 =m 1v 11 2+(m 1v 10+m 2v 20) 2m 2−2m 1(m 1v 10+m 2v 20)v 11m 2+m 1 2m 2v 11 2\begin{aligned}m_1 v_{10}^2 + m_2 v_{20}^2 = m_1 v_{11}^2 + m_2 (\frac{m_1 v_{10} + m_2 v_{20} - m_1 v_{11}}{m_2})^2\\= m_1 v_{11}^2 + m_2 ((\frac{m_1 v_{10} + m_2 v_{20}}{m_2})^2- 2\frac{m_1(m_1 v_{10} + m_2 v_{20}) v_{11}}{m_2^2}+ \frac{m_1^2}{m_2^2} v_{11}^2\\=m_1 v_{11}^2 + \frac{(m_1 v_{10} + m_2 v_{20})^2}{m_2} - \frac{2 m_1 (m_1 v_{10} + m_2 v_{20}) v_{11}}{m_2} + \frac{m_1^2}{m_2} v_{11}^2\end{aligned}
m 1v 10 2+m 2v 20 2−(m 1v 10+m 2v 20) 2m 2=(m 1+m 1 2m 2)v 11 2−2m 1(m 1v 10+m 2v 20)v 11m 2\begin{aligned}m_1 v_{10}^2 + m_2 v_{20}^2 - \frac{(m_1 v_{10} + m_2 v_{20})^2}{m_2} = (m_1 + \frac{m_1^2}{m_2}) v_{11}^2 - \frac{2 m_1 (m_1 v_{10} + m_2 v_{20}) v_{11}}{m_2} \end{aligned}
Define k=m 1m 2k=\frac{m_1}{m_2}
kv 10 2+v 20 2−(kv 10+v 20) 2 =(k+k 2)v 11 2−2k(kv 10+v 20)v 11 kv 10 2−kv 10(kv 10+2v 20) =(k+k 2)v 11 2−2k(kv 10+v 20)v 11 (k−k 2)v 10 2−2kv 10v 20 =(k+k 2)v 11 2−2k(kv 10+v 20)v 11 (1−k)v 10 2−2v 10v 20 =(1+k)v 11 2−2(kv 10+v 20)v 11 \begin{aligned}k v_{10}^2 + v_{20}^2 - (k v_{10} + v_{20})^2 &= (k + k^2) v_{11}^2 - 2 k (k v_{10} + v_{20}) v_{11}\\k v_{10}^2 - k v_{10} (k v_{10} + 2 v_{20}) &= (k + k^2) v_{11}^2 - 2 k (k v_{10} + v_{20}) v_{11}\\( k - k^2 ) v_{10}^2 - 2 k v_{10} v_{20} &= (k + k^2) v_{11}^2 - 2 k (k v_{10} + v_{20}) v_{11}\\( 1 - k ) v_{10}^2 - 2 v_{10} v_{20} &= (1+k) v_{11}^2 - 2(k v_{10} + v_{20}) v_{11}\\\end{aligned}
So, Case 1
v 11 =kv 10+v 20+v 10−v 201+k=v 10 v 21 =m 1v 10+m 2v 20−m 1v 11m 2=v 20\begin{aligned}v_{11} &= \frac{2 (k v_{10} + v_{20}) ± \sqrt{4 (k v_{10} + v_{20})^2 + 4 (1 + k) ( ( 1 - k ) v_{10}^2 - 2 v_{10} v_{20} )} }{2 ( 1 + k )}\\&= \frac{k v_{10} + v_{20} ± \sqrt{(k v_{10} + v_{20})^2 +( 1 - k^2 ) v_{10}^2 - 2 (1 + k) v_{10} v_{20}} }{1 + k }\\&= \frac{k v_{10} + v_{20} ± \sqrt{( v_{10} - v_{20} )^2}}{1 + k} \end{aligne\begin{aligned}v_{11} &= \frac{k v_{10} + v_{20} + v_{10} - v_{20} }{1 + k} = v_{10} \\v_{21} &= \frac{m_1 v_{10} + m_2 v_{20} - m_1 v_{11}} {m_2} = v_{20}\end{aligned}
which is the trivial solution for same velocity of both bodies.
Case 2
v 11= kv 10+v 20−v 10+v 201+k = k−11+kv 10+21+kv 20 v 21= m 1v 10+m 2v 20−m 1v 11m 2 = kv 10+v 20−kv 11 = kv 10+v 20−k(k−1)1+kv 10−2k1+kv 20 = 2k1+kv 10+1−k1+kv 20\begin{aligned}v_{11} =& \frac{k v_{10} + v_{20} - v_{10} + v_{20} }{1 + k} \\=&\frac{k - 1}{1 + k} v_{10} + \frac{2}{1 + k} v_{20} \\v_{21} =& \frac{m_1 v_{10} + m_2 v_{20} - m_1 v_{11}} {m_2}\\=& k v_{10} + v_{20} - k v_{11}\\=& k v_{10} + v_{20} - \frac{k(k - 1)}{1 + k} v_{10} - \frac{2 k}{1 + k} v_{20}\\=& \frac{2 k}{1 + k} v_{10} + \frac{1 - k}{1 + k} v_{20}\end{aligned}
Some special situations.
Situation 1. body 2 is at rest and m 1m_1 is so small than m 2m_2 meaning nearly zero kk , then
v 11=−v 10 v 21=0\begin{aligned} v_{11} = - v_{10} \\ v_{21} = 0\end{aligned}
Situation 2. m 1m_1 is equal to m 2m_2 meaning k=1k=1 , then
v 11=v 20 v 21=v 10\begin{aligned} v_{11} = v_{20} \\ v_{21} = v_{10}\end{aligned}
